Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}9x-3y &= -6 \\ -7x+6y &= 7\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $1$ $\begin{align*}18x-6y &= -12\\ -7x+6y &= 7\end{align*}$ Add the top and bottom equations. $11x = -5$ Divide both sides by $11$ and reduce as necessary. $x = -\dfrac{5}{11}$ Substitute $-\dfrac{5}{11}$ for $x$ in the top equation. $9( -\dfrac{5}{11})-3y = -6$ $-\dfrac{45}{11}-3y = -6$ $-3y = -\dfrac{21}{11}$ $y = \dfrac{7}{11}$ The solution is $\enspace x = -\dfrac{5}{11}, \enspace y = \dfrac{7}{11}$.